Spreading verticies out function?
Assuming you are using Blender 2.8, just follow the tutorial until the desired vertices are selected, then do the following:


Press Ctrl+B (to activate the Bevel Tool)
Press V (to make the tool act on vertices)
Drag your mouse outwards until you get the same shape as the one in the tutorial.


Note that, whenever you've selected a tool (such as the bevel tool), some options related to its use get listed on the status bar just below the 3D Viewport.